Cp-Cpk-formulasThis chapter is a detailed discussion of the basics of data, variation and Statistical Control. You will learn here how to understand a distribution graph, Understand what process capability is and to enumerate the various methods and indices used to calculate the process capability.

Data Facts – Basic nature of data

Two kinds of data are normally found in any organization:

1. Continuous Data

Data which can be measured and which characterizes a product or some process features in terms of its size, weight and volts. This type of data is said to be continuous by nature. In other words, the measurement scale can be meaningfully divided into finer and finer increments using the concept of least count.

2. Discrete Data

When data cannot be measured but only observed and when it occurs in the form of frequency of occurrences; e.g., the number of times some events happen or fail to happen. For example, the number of typing errors or number of days an employee is absent cannot be measured but can be counted.

The validity of inferences made from discrete data depends a lot upon the number of observations. The more the number of observations the more would be the accuracy of inference. Thus, a sample size required to characterize discrete data would be typically much larger than that required when continuous data is used.

Data Facts – Location and Variation Estimates

Location and variation quantify essential information about output of the process. Location is the most central part of the process. For example, bolts manufactured may vary in their thickness. Location refers to that value which represents the setting of the process. It could be the mean of all the values; could be the median or the mode. Of these, mean is the most commonly used measure. Variability is the spread or range of the process where most values occur. Measures of variation can be either Standard Deviation or Range.


The need for quality control arises from the fact that even after the quality standards have been specified, variation in quality is unavoidable due to process variation.

For example

A machine is producing 100,000 bolts per day of 2-inch lengths each. It is very unlikely that all bolts measure exactly 2 inches. If the measuring instrument is sufficiently good, we can detect some bolts that are slightly less than 2 inches and some that are slightly more than 2 inches. This leads to a search for the possible causes of variation in the product.

A measure of variation or dispersion is one that measures the extent to which there are differences between individual observations and some central or average value.

Note that in measure of variation we would be interested in the amount of variation or its degree and not in the direction. For example, a measure of 4 inches below the mean has just as much variation as a measure of six inches above the mean.

Types of variation

The variation of a quality characteristic can be divided under two heads:

1. Chance variation
2. Assignable variation

Chance variation

These are variations that result from many minor causes. These causes behave in a random manner. This variation follows statistical property like Normal distribution,
and hence, we say it is predictable variation. This type of variation is permissible and actually inevitable in a practical manufacturing environment. There is no way in which it can completely be eliminated – when the variability present in a production process is confined to chance variation only, the process is said to be in a state of statistical control.

Assignable variation

These variations may be attributed to special non-random causes. This variation does not follow any statistical law and hence, we can say it is unpredictable. Such variations can be a result of several factors such as a change in the raw material, a new operation, an improper machine setting, broken or worn-out parts, mechanical faults in plants etc.

The value of Quality control lies in the ability to detect these variations in a process. In fact, these variations can be discovered before the product becomes defective. Understanding Distribution curve. Distribution can differ in:

1. Spread: The distribution curve 1 represents two distributions with same mean but with different dispersions.

2. Location: The distribution Curve 2 has two distributions that have the same dispersion but with unequal means X1 and X2.

3. Shape: The two distributions have unequal dispersion.

Process Capability

Process Capability represents the best performance (i.e. minimum spread) of the process, when the process is operating in a state of Statistical control due to no assignable causes.

It indicates only the natural fluctuations that take place in Key characteristic of a process. When a process is in a state of statistical control the only variations in the process are due to the chance causes. Thus Capability is determined by the variation that comes from chance causes.

Mathematically, Capability = Six Sigma calculated from a set of individual measurements.

Potential capability (Cp).

This is the simplest and most straightforward indicator of process capability. It is defined as the ratio of the specification range to the process range; using ± 3 sigma limits we can express this index as:

Cp = (USL-LSL)/(6*Sigma)

Put into words, this ratio expresses the proportion of the range of the normal curve that falls within the engineering specification limits.

Capability Ratio (Cr)

This index is equivalent to Cp; specifically, it is computed as 1/Cp
A drawback of the Cp index is that it really evaluates only process spread and ignores the process average. If the system is not centered at the middle of the specifications, the Cp index may be misleading.

Assuming the system is centered, a Cp value of 1 indicates that the system is producing 99.73% of output within specification limits.
Lower/upper potential capability (Cpl, Cpu.)

A major shortcoming of the Cp (and Cr) index is that it may yield erroneous information if the process is not on target, that is, if it is not centered or if the process shows one-sided specifications such as minimum length, etc.

For non-centered distribution (such as in the diagram shown) upper and lower potential capability indices can be computed which would aptly reflect the deviation of the observed process mean from the LSL and USL.

Demonstrated Excellence

It is denoted by Cpk and takes the value of either Cpl or Cpu based on which value is lower.
Cpk = Smaller value of [(USL – Mean)/ 3 Sigma, (Mean – LSL)/3 Sigma]

Unlike potential capability, demonstrative excellence takes into account the centering of the key characteristics of the process.

While Cp is a function of the range, Cpk is the function of the average i.e. the mean.

The Cpk tells how well a system can meet specification limits while accounting for the location of the average. The Cpk index modifies the Cp index to account for the location of the average (or center). A Cpk of 1 indicates that the system is producing at least 99.73% within the specification limits.

Statistic Definition of Six Sigma

We know that process outputs vary. This variation follows some pattern known as Distribution.

Tchebyeff’s Theorem states that: “No matter what the shape of the distribution is, at least 75 percent of the values will fall within ± 2 standard deviations from the mean of the distribution, and at least 89 percent of the values will lie within ± 3 standard deviations from the mean.”
For a normal distribution, the following relationships hold good:

Mean ± 1-sigma covers 68.27% of the items
Mean ± 2 sigma covers 68.45% of the items
Mean ± 3 sigma covers 99.73% of the items

Most output processes have output that follows a normal distribution as shown by curve X in the diagram. A process that is naturally centered at O will have a natural spread around O of plus or minus three-sigma standard deviation.

In the case of Six Sigma, this process variation is only half the width of the design tolerances for the process, that is to say, the difference between the upper specification limit (USL) and lower specification limit (LSL). Since, 99.9973 per cent of the process output is contained by this natural spread, a process running at O is highly capable of meeting the design specifications and only 0.002 defects per million opportunities will arise since only 0.002 parts per million are outside this curve.

Six Sigma and Process Capability

We have seen that all processes will have some variation. In a stable process, this variation will be equal to plus or Minus 3 Sigma from its own average.
This plus or minus 3 Sigma (six Sigma) is called Process Capability.

If process capability is less that the tolerance or expectations then the process will produce lesser defects.

Six Sigma and Defect per million opportunities
Six Sigma brings about process improvement by reducing Defects per million opportunities in the process.

Process Shift Concept

Regulating processes so that they always remain on target may not be feasible in the long term. In practical scenarios the process is likely to deviate from its natural centered position by approximately one and a half standard deviations.
Under these circumstances, one side will be 7.5 Sigma and the other side will be 4.5 Sigma.
Under Six Sigma we focus on the long-term capability, which means that we have to account for a 1.5 Sigma shift in the process average.

* Mean ± 1 sigma covers 68.27% of the items
* Mean ± 2 sigma covers 68.45% of the items
* Mean ± 3 sigma covers 99.73% of the items
* if process is operating well; 3.4 million effects per million opportunities
* Process Capability: Variation seen in a stable process
* Process average shift in real life is 1.5 over any period of time
* The Higher the sigma level, the more accurate the process

1. Data can be continuous or discrete.
2. A set of data can be measured for its location and spread.
3. Location is measured using mean, median or mode.
4. Spread of a data is determined using either Standard deviation or range.
5. Measure of dispersion measures the extent to which there are differences between individual observations and some central or average value.
6. Variations can be chance or assignable.
7. Chance variations are caused by random causes that cannot be completely removed form the process.
8. Assignable variations may be attributed to special non-random causes and these should be corrected.
9. When process shows only chance variation, it is under statistical control.
10. Process Capability represents the best performance (i.e. minimum spread) of operating in a state of Statistical control.
11. Potential capability is the ratio that expresses the proportion of the range of the normal curve that falls within the engineering specification limits.
12. For non-centered distribution, upper and lower potential capability indices are computed

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